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Register a new userNash bargaining with a nondeterministic threat
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Added by
Kerry Soileau
Publication date
2008
fields
Informatics Engineering
and research's language is
English
Authors
Kerry Michael Soileau
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We consider bargaining problems which involve two participants, with a nonempty closed, bounded convex bargaining set of points in the real plane representing all realizable bargains. We also assume that there is no definite threat or disagreement point which will provide the default bargain if the players cannot agree on some point in the bargaining set. However, there is a nondeterministic threat: if the players fail to agree on a bargain, one of them will be chosen at random with equal probability, and that chosen player will select any realizable bargain as the solution, subject to a reasonable restriction.
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This paper addresses the paucity of models of matching markets, both one-sided and two-sided, when utility functions of agents are cardinal. The classical Hylland-Zeckhauser scheme cite{hylland}, which is the most prominent such model in economics, can be viewed as corresponding to the linear Fisher model, which is most elementary model in market equilibria. Although HZ is based on the attractive idea of using a pricing mechanism, from the viewpoint of use in applications, it has a serious drawback, namely lack of computational efficiency, due to which solving instances of size even 4 or 5 is difficult. We propose a variety of Nash-bargaining-based models, several of which draw from general equilibrium theory, which has defined a rich collection of market models that generalize the linear Fisher model in order to address more specialized and realistic situations. The Nash bargaining solution satisfies Pareto optimality and symmetry and the allocations it yields are remarkably fair. Furthermore, since the solution is captured via a convex program, it is polynomial time computable. In order to be used in industrial grade applications, we give implementations for these models that are extremely time efficient, solving large instances, with $n = 2000$, in one hour on a PC, even for a two-sided matching market. The idea underlying our work has its origins in Vazirani (2012), which viewed the linear case of the Arrow-Debreu market model as a Nash bargaining game and gave a combinatorial, polynomial time algorithm for finding allocations via this solution concept, rather than the usual approach of using a pricing mechanism.
This paper is an attempt to deal with the recent realization (Vazirani, Yannakakis 2021) that the Hylland-Zeckhauser mechanism, which has remained a classic in economics for one-sided matching markets, is likely to be highly intractable. HZ uses the power of a pricing mechanism, which has endowed it with nice game-theoretic properties. Hosseini and Vazirani (2021) define a rich collection of Nash-bargaining-based models for one-sided and two-sided matching markets, in both Fisher and Arrow-Debreu settings, together with implementations using available solvers, and very encouraging experimental results. This naturally raises the question of finding efficient combinatorial algorithms for these models. In this paper, we give efficient combinatorial algorithms based on the techniques of multiplicative weights update (MWU) and conditional gradient descent (CGD) for several one-sided and two-sided models defined in HV 2021. Additionally, we define for the first time a Nash-bargaining-based model for non-bipartite matching markets and solve it using CGD. Furthermore, in every case, we study not only the Fisher but also the Arrow-Debreu version; the latter is also called the exchange version. We give natural applications for each model studied. These models inherit the game-theoretic and computational properties of Nash bargaining. We also establish a deep connection between HZ and the Nash-bargaining-based models, thereby confirming that the alternative to HZ proposed in HV 2021 is a principled one.
We study decentralized markets with the presence of middlemen, modeled by a non-cooperative bargaining game in trading networks. Our goal is to investigate how the network structure of the market and the role of middlemen influence the markets efficiency and fairness. We introduce the concept of limit stationary equilibrium in a general trading network and use it to analyze how competition among middlemen is influenced by the network structure, how endogenous delay emerges in trade and how surplus is shared between producers and consumers.
Shapleys impossibility result indicates that the two-person bargaining problem has no non-trivial ordinal solution with the traditional game-theoretic bargaining model. Although the result is no longer true for bargaining problems with more than two agents, none of the well known bargaining solutions are ordinal. Searching for meaningful ordinal solutions, especially for the bilateral bargaining problem, has been a challenging issue in bargaining theory for more than three decades. This paper proposes a logic-based ordinal solution to the bilateral bargaining problem. We argue that if a bargaining problem is modeled in terms of the logical relation of players physical negotiation items, a meaningful bargaining solution can be constructed based on the ordinal structure of bargainers preferences. We represent bargainers demands in propositional logic and bargainers preferences over their demands in total preorder. We show that the solution satisfies most desirable logical properties, such as individual rationality (logical version), consistency, collective rationality as well as a few typical game-theoretic properties, such as weak Pareto optimality and contraction invariance. In addition, if all players demand sets are logically closed, the solution satisfies a fixed-point condition, which says that the outcome of a negotiation is the result of mutual belief revision. Finally, we define various decision problems in relation to our bargaining model and study their computational complexity.
Bargaining networks model the behavior of a set of players that need to reach pairwise agreements for making profits. Nash bargaining solutions are special outcomes of such games that are both stable and balanced. Kleinberg and Tardos proved a sharp algorithmic characterization of such outcomes, but left open the problem of how the actual bargaining process converges to them. A partial answer was provided by Azar et al. who proposed a distributed algorithm for constructing Nash bargaining solutions, but without polynomial bounds on its convergence rate. In this paper, we introduce a simple and natural model for this process, and study its convergence rate to Nash bargaining solutions. At each time step, each player proposes a deal to each of her neighbors. The proposal consists of a share of the potential profit in case of agreement. The share is chosen to be balanced in Nashs sense as far as this is feasible (with respect to the current best alternatives for both players). We prove that, whenever the Nash bargaining solution is unique (and satisfies a positive gap condition) this dynamics converges to it in polynomial time. Our analysis is based on an approximate decoupling phenomenon between the dynamics on different substructures of the network. This approach may be of general interest for the analysis of local algorithms on networks.
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